Convective nonlocal Cahn-Hilliard equations with reaction terms

TL;DR

This paper introduces and analyzes nonlocal Cahn-Hilliard equations with reaction terms, establishing well-posedness and long-term behavior such as bounded absorbing sets and global attractors.

Contribution

It extends the classical Cahn-Hilliard framework to nonlocal interactions with reaction terms, providing rigorous analysis of existence, uniqueness, and dynamical properties.

Findings

Proved well-posedness of the nonlocal Cahn-Hilliard equations.

Established existence of bounded absorbing sets.

Demonstrated the existence of global attractors.

Abstract

We introduce and analyze the nonlocal variants of two Cahn-Hilliard type equations with reaction terms. The first one is the so-called Cahn-Hilliard-Oono equation which models, for instance, pattern formation in diblock-copolymers as well as in binary alloys with induced reaction and type-I superconductors. The second one is the Cahn-Hilliard type equation introduced by Bertozzi et al. to describe image inpainting. Here we take a free energy functional which accounts for nonlocal interactions. Our choice is motivated by the work of Giacomin and Lebowitz who showed that the rigorous physical derivation of the Cahn-Hilliard equation leads to consider nonlocal functionals. The equations also have a transport term with a given velocity field and are subject to a homogenous Neumann boundary condition for the chemical potential, i.e., the first variation of the free energy functional. We first establish the well-posedness of the corresponding initial and boundary value problems in a weak setting. Then we consider such problems as dynamical systems and we show that they have bounded absorbing sets and global attractors.